If D is close to two (2), then the model is likely free of autocorrelation.

If D is close to four (4), then negative autocorrelation is probably present.
A rule of thumb is that test statistic values in the range of 1.5 to 2.5 are relatively normal. Values outside of this range could be cause for concern. The test statistic is compared to lower and upper critical values which are DL and DU for specific level of significance ? to test for the autocorrelation. Furthermore, DL are Durbin Watson lower control limit whereas DU are Durbin Watson upper control limit.
The hypothesis testing for positive autocorrelation of the Durbin Watson test are
H0 : ? = 0
H1: ? > 0
This is the criteria for positive correlation
If D < DL rejects H0 If D > DU do not reject H0 If DL <D< DU the test is inconclusive
The hypothesis testing for negative autocorrelation of the Durbin Watson test are
H0 : ? = 0
H1: ? > 0
This is the criteria for negative correlation
If 4-D < DL reject H0 If 4-D > DU do not reject H0 If DL < 4-D < DU the test is inclusive
3.4 Stationary
A time series has a stationarity if a shift in time does not cause a change in the shape of the distribution. Basic properties of the distribution like the mean, variance and covariance are constant over the time. Most forecasting methods assume that a distribution has stationarity. For example, auto-covariance and autocorrelation rely on the assumption of stationarity. It is hard to tell whether the model is stationary or not. Thus, if we are not sure about the stationarity of the model, several testing can be done such as Unit root test ADDIN CSL_CITATION { “citationItems” : { “id” : “ITEM-1”, “itemData” : { “DOI” : “10.5093/cl2010v21n1a6”, “ISSN” : “1130-5274”, “abstract” : “The theme of unit roots in macroeconomic time series have received a great amount of attention in terms of theoretical and applied research over the last three decades. Since the seminal work by Nelson and Plosser (1982), testing for the presence of a unit root in the time series data has become a topic of great concern. This issue gained further momentum with Perron’s 1989 paper which emphasized the importance of structural breaks when testing for unit root processes. This paper reviews the available literature on unit root tests taking into account possible structural breaks. An important distinction between testing for breaks when the break date is known or exogenous and when the break date is endogenously determined is explained. We also describe tests for both single and multiple breaks. Additionally, the paper provides a survey of the empirical studies and an application in order for readers to be able to grasp the underlying problems that time series with structural breaks are currently facing.”, “author” : { “dropping-particle” : “”, “family” : “Glynn”, “given” : “John”, “non-dropping-particle” : “”, “parse-names” : false, “suffix” : “” }, { “dropping-particle” : “”, “family” : “Perera”, “given” : “Nelson”, “non-dropping-particle” : “”, “parse-names” : false, “suffix” : “” }, { “dropping-particle” : “”, “family” : “Verma”, “given” : “Reetu”, “non-dropping-particle” : “”, “parse-names” : false, “suffix” : “” } , “container-title” : “Journal of Quantitative Methods for Economics and Business Administration”, “id” : “ITEM-1”, “issue” : “1”, “issued” : { “date-parts” : “2007” }, “page” : “63-79”, “title” : “Unit Root Tests and Structural Breaks: A Survey with Applications”, “type” : “article-journal”, “volume” : “3” }, “uris” : “http://www.mendeley.com/documents/?uuid=5ede21ae-6a9d-4a2b-8cf4-d297fd8071f3” } , “mendeley” : { “formattedCitation” : “(Glynn, Perera, and Verma 2007)”, “plainTextFormattedCitation” : “(Glynn, Perera, and Verma 2007)”, “previouslyFormattedCitation” : “(Glynn, Perera, and Verma 2007)” }, “properties” : { “noteIndex” : 0 }, “schema” : “https://github.com/citation-style-language/schema/raw/master/csl-citation.json” }(Glynn, Perera, and Verma 2007), KPSS test, a run sequence plot, The Priestley-Subba Rao (PSR) Test or Wavelet-Based Test.

3.5 Box Jenkins Method
The Box-Jenkins method is a time series analysis, forecasting and can be used in many areas and situation which involve in choosing a suitable model. The Box-Jenkins method is one of the most popular time series forecasting methods in business and economics. The method uses a systematic procedure to select an appropriate model, namely, Integrated Autoregressive Moving Average (ARIMA) models. Following Johnson (1997), the general notation for the order of a seasonal ARIMA model with both seasonal and non-seasonal factors is ARIMA(p,d,q)×(P,D,Q), and the term (p,d,q) gives the order of the non-seasonal part of the ARIMA model, the term (P,D,Q) gives the order of the seasonal part. A general ARIMA model has the following form (Bowerman and O’Connell, 1993 ).

By using the following process, the Box-Jenkins method can be carried out. The first step in Box Jenkins is Model identification. By using Historical data, it can be used to identify appropriate Box Jenkins model. Firstly, time series plotting can be used to check whether there is a liner trend, stationarity, outliers, seasonal pattern and others in the time series, as well as the mean of the time series is constant or not. Then, the natural logarithmic transformation is applied to stabilise the variance if the mean of the time series is not relatively constant over time.

Table 3.1 summary of transformation based on the value of ?
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Power Transformation
1 “raw”
-2 Reciprocal square
-1 Reciprocal
-0.5 Reciprocal root
0 Logarithm
0.5 Square root
;1 Square
After that, if the mean is still not stationary, differencing can be applied to transform it into stationary if it is not stationary. Differencing is one of the way that can make time series stationary. It also help to stabilize the mean of a time series by removing changes in the level of a time series, and so eliminating or reducing trend and seasonality.The first differencing operater, defined by
yt’=yt -yt-1=1-Byt (3.5)
where
B= backward shift operator
Sometimes, the differenced data are not appear stationary so it is need to difference the data second time in order to obtain stationary series. The second differencing is
= yt’ – yt-1′ =(yt-yt-1)-(yt-1-yt-2)
=yt-2yt-1+yt-2 =(1-B)²yt =(1-2B+B²) (3.6)
When the time series has seasonal component, a seasonal differencing can be used. A seasonal difference is the difference between an observation and the corresponding observation from the previous year. So,
yt’= yt-yt-m (3.7)
where m=number of seasons.

These are also called “lag-m differences” as we subtract the observation after a lag of m periods.
When the time series is in a stationary condition, the model order of autoregressive (AR) compnent and moving average (MA) component can be determined by using graphical plot of autocorrelation function (ACF) and partial autocorrelation function (PACF) . Autoregression model used a linear combination of past values of the variable. The autoregressive model of order p can be written as
yt=c+?1+yt-1+?2yt-2+…+?pyt-p+et (3.8)
where et = white noise
?= coefficient
It is AR(p) model. Rather than use past values of the forecast variable in a regression, a moving average model uses past forecast errors in a regression-like model. Then, the moving average model of order q can be written as
yt= c+et+?1et-1+?2et-2+…+?qet-q (3.9)
where et is a white noise. This is a MA (q) model.

The value of p can be determined from the partial autocorrelations function (PACF). If the ACF exponential decay and PACF cuts off, the model suggested the AR term. The value of q can be determined by autocorrelation function (ACF). If the ACF cut offs and PACF exponential decay, the model suggested the MA term. If the ACF and PACF shows the exponential delay, then the model is ARMA process. The behaviour of ACF and PACF for stationary are summarized in Table 3.2
Table 3.2 The summary behaviour of ACF and PACF for stationary
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Model ACF PACF
MA(q): moving average of order q Cut off after lag q Dies down
AR(p): autoregressive of order p Dies down Cuts off after lag p
ARMA(p,q):mixed autoregressive moving average of order (p , q) Dies down Dies down
AR(p) or MA(q) Cuts off after lag q Cuts off after lag p
No order AR or MA (White Noise or Random process) No spike No spike
The full model can be written as
yt’= c+ ?1 +…+ yt-1′ ?p + …+ yt-p’+?1et-1+…+?qet-q+et (3.10)
where
yt’ = differenced series
We can call this an ARIMA (p,d,q) model, where
p = order of the autoregressive part
d = degree of first differencing involved
q = order of moving average part
Next, the general seasonal ARIMA model of orders (p,d,q)x(P,D,Q) with period d is
(1-?pB)(1-?pBm) (1-B)(1-Bd)yt = (1-?qB)(1+?Q Bm)et (3.11)
where
m = number of observations per year
?pB = seasonal autoregressive operator for non-seasonal part of model
?pB= seasonal autoregressive operator for seasonal part of model
?qB = seasonal moving average operator for non-seasonal part of model
?QB= seasonal moving average operator for seasonal part of model
During a second stage which is estimation stage, estimate the model coefficients by selecting the best-fit model based on the smallest values of AIC and SIC tests. Furthermore, in order to check the adequacy of the estimated model diagnostic checking is carried out and if need to, alternative models may be considered. By using ACF and PACF residuals, it can verify the “white noise” characteristics of the residual series from the selected model when the ACF and PACF residuals is within the 0.05 significance level.
Next, Ljung–Box Chi-Square statistic can be used as a diagnostic tool to test the lack of fit of a time series model (Ljung ; Box, 1978). Ljung–Box Chi-Square statistic is one of the way to assess if the residual from the Box Jenkins model follow the assumptions. Hypothesis testing for the Ljung–Box Chi-Square statistic is:
H0 : The model is adequate
H1 : The model is inadequate
If the p-value of the Ljung-Box Chi Square statistic is small (say, p-value;0.05), the null hypothesis are rejected thus the selected model is considered inadequate, and then a modified model will be established until a satisfactory model can be determined.

Next, forecast can be calculated. The main purpose of fitting ARMA schemes is to project the series forward beyond the sample period or out of sample. It should be noted that, in all that follows we will assume that observations are only available for periods 1 to n, and that all forecasts are made conditional on information available at time n. We look at the residuals to determine how accurate the model predicts. The desired accuracy of the forecasts depends on the analyst’s goal.

3.6 Mean Absolute Percent Error (MAPE)
The MAPE (Mean Absolute Percent Error) measures the size of the error in percentage terms. It is calculated as the average of the unsigned percentage error. Percentage errors have the advantage of being scale-independent, and so are frequently used to compare forecast performance between different data sets.
MAPE= (1n|Actual-Forecast||Actual|)*100 (3.12)
where
n= number of predicted values
The smaller the value of MAPE, the more accurate the forecast. The judgement of forecast accuracy based on MAPE value was summarized in the Table 3.3 below
Table 3.3 The judgement of forecast accuracy based on MAPE value
MAPE Judgement of forecast accuracy
Less than 10% Highly accurate
11% to 20% Good forecast
21% to 50% Reasonable forecast
More than 51% Inaccurate forecast
3.7 Mean Square Error (MSE)
The Mean Square Error (MSE) is a widely used criterion for the choice of a forecasting performance rule. The minimum the value of MSE, the more accurate the forecast. Mean Square Error (MSE) is a measure of dispersion of forecast errors by taken the average of the squared individual errors.
Formula:
MSE = (actual-forecast)²n (3.13)
n= number of predicted values
CHAPTER 4
EXPECTED RESULT
4.1 Expected result
In this study, our objective is to study the behaviour of the Air Asia passenger data by identify whether there is a trend or pattern by using time series plot. By using the result obtained from the time series plot, we can determine whether the data have trend, seasonality or cyclic behavior that can be seen clearly from the output.
Moreover, we need to apply ARIMA model in Air Asia passenger data from January 2008 until Ogos 2012 in order to find the best forecasting model. We are expecting the best model will be determined which is seasonal ARIMA model.
Furthermore, we need to forecast the future of Air Asia passenger for January 2008 until Ogos 2012 by using Box Jenkins Method so that the researchers can obtain a wide knowledge regarding Air Asia passenger data besides other literature review on Air Asia passenger that we can make it as a reference. Based on the result, the trend of the Air Asia passenger can be identified in the future.
Last but not least, we have to evaluate the forecasting performance of ARIMA model. The best forecasting model will be determined and chosen from the forecasting accuracy measure which is MAPE and MSE. The forecasting model with the most minimum forecasting error will be selected as the best forecasting model.
David, J. R. (2011). Budget airlines. Retrieved November 11, 2011, from http://blog.malaysia-asia.my/2011/08/budget-airlines.html
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Mohd, Norhaidah et al. 2013. “Time Series Behaviour of the Number of Air Asia Passengers?: A Distributional Approach.” (February).

O’Connell, John F., and George Williams. 2005. “Passengers’ Perceptions of Low Cost Airlines and Full Service Carriers: A Case Study Involving Ryanair, Aer Lingus, Air Asia and Malaysia Airlines.” Journal of Air Transport Management 11(4): 259–72.